Question

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length 1.00 $\times$ 10${}^{-14}$ m, the approximate diameter of a nucleus?

Short answer

The energy difference $\mathrm{\Delta }E=2.42\times {10}^{-13}$ J.

Step-by-step solution

Understand the system

The problem involves a proton confined in a cubical box, a situation where quantum mechanics is applicable. We will use the particle in a box model to calculate energy levels.

Know the Energy Formula for a Cubical Box

The energy levels for a particle in a 3D box are given by$$E_{n_x,n_y,n_z}=\frac{h^2}{8m{L}^2}(n_x^2+n_y^2+n_z^2)$$where $h$ is Planck's constant, $m$ is the mass of the proton, $L$ is the side of the box, and $n_x,n_y,n_z$ are quantum numbers (1, 2, 3,...).

Set Quantum Numbers for Two Lowest Energy Levels

The lowest energy state is the ground state with $n_x=n_y=n_z=1$. The next energy state involves exciting one quantum number by an integer; often it's $n_x=2,n_y=n_z=1$.

Calculate Ground State Energy

For the ground state, where $n_x=n_y=n_z=1$, the energy is:$$E_{1,1,1}=\frac{h^2}{8m{L}^2}(1^2+1^2+1^2)=\frac{3h^2}{8m{L}^2}$$

Calculate First Excited State Energy

For the first excited state, a potential scenario is $n_x=2,n_y=n_z=1$, giving:$$E_{2,1,1}=\frac{h^2}{8m{L}^2}(2^2+1^2+1^2)=\frac{6h^2}{8m{L}^2}$$

Compute Energy Difference

The energy difference between the first excited state and the ground state is:$$\mathrm{\Delta }E=E_{2,1,1}-E_{1,1,1}=\left(\frac{6h^2}{8m{L}^2}\right)-\left(\frac{3h^2}{8m{L}^2}\right)=\frac{3h^2}{8m{L}^2}$$

Insert Values to Find Numerical Result

Insert the values for Planck's constant $h=6.626\times {10}^{-34}\text{Js}$, the mass of the proton $m=1.673\times {10}^{-27}\text{kg}$, and the side length $L=1.00\times {10}^{-14}\text{m}$:$$\mathrm{\Delta }E=\frac{3\times (6.626\times {10}^{-34}{)}^2}{8\times 1.673\times {10}^{-27}\times (1.00\times {10}^{-14}{)}^2}$$

Calculate Final Numerical Solution

Carry out the computation in step 7 to find $\mathrm{\Delta }E$.

Key concepts

Particle in a Box

The "particle in a box" is a popular model in quantum mechanics. It simplifies the complex nature of particles confined in a limited space, like the nucleus of an atom. Think of it as a way to visualize how particles behave when trapped in a very tiny space.

In this model, the box is usually considered to have infinitely high walls, meaning the particle cannot escape. It's a useful concept because it allows us to calculate the possible energy levels of the particle based on their quantized nature. In our specific problem, a proton is considered to be in such a box, which represents the confined space of a nucleus. The cubical shape we use is idealized to make the math easier, but it captures the essential physics.

Energy Levels

Energy levels in quantum mechanics are like "rungs on a ladder." A particle can only exist on these specific steps and nowhere in between.

When we talk about energy levels in the "particle in a box" model, these levels are determined numerically using an equation. This equation includes quantum numbers, Planck's constant, the mass of the particle, and the size of the confinement region (like our cubical box). Different arrangements of quantum numbers create different energy levels.

• The ground state is the lowest energy level, where the quantum numbers are all minimized (e.g., 1, 1, 1).
• Excited states are higher energy levels where at least one quantum number is increased.

This is how we derive energy levels for our proton in the box.

Planck's Constant

Planck's constant is a fundamental constant in physics that plays an essential role in quantum mechanics. It defines the size of the quantization found in nature.

In the formula for the "particle in a box," Planck's constant appears as a squared value. This constant, denoted as $h$, is incredibly small, approximately $6.626\times {10}^{-34}\text{Js}$. Its tiny size indicates that the energy differences we're discussing are extremely small at the quantum level.

Planck's constant essentially "scales" the quantum mechanical world to its microscopic nature, affecting how we calculate energy levels and the precise nature of quantum jumps between these levels.

Quantum Numbers

Quantum numbers are key to understanding the quantized nature of particles in quantum mechanics. In the "particle in a box" model, they help define the possible energy levels.

There are typically three quantum numbers $n_x$, $n_y$, and $n_z$ in a three-dimensional box. Each of these numbers can be an integer starting from 1.

• The smallest numbers represent the lowest energy state, or ground state.
• Higher numbers indicate excited states.

These values are crucial for calculating the energy levels using the formula provided. By changing these numbers, you compute different energy scenarios for the particle, such as moving from ground state to an excited state, which is what causes energy differences as per our calculations.